Higher order symmetries and integrating factors for ordinary differential equations

Symmetry analysis and conservation laws are widely used in analyzing and solving differential equations. Conservation laws are also called first integrals when dealing with ordinary differential equations (ODEs). We explore the complementary nature of symmetry analysis and conservation laws; specifically, the use of symmetries to find integrating factors and, conversely, the use of conservation laws to seek new symmetries. In particular, we show two new fundamental results. Firstly, we show that a higher-order symmetry of an ODE induces a Lie point symmetry of the corresponding integrating factor determining equations (IFDEs). Moreover, we obtain an explicit expression for the infinitesimal generator of this induced point symmetry, enabling one to find new integrating factors of existing ODEs and thus possibly to solve a wider range of ODEs. Secondly, we show that the converse also holds, i.e. a point symmetry of the IFDEs for a given ODE induces a higher-order symmetry of the ODE.